If $\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{5 \pi}{15} \cos \frac{7 \pi}{15} \cos \frac{30 \pi}{15} = x$,then $\frac{1}{8x} =$

The value of the expression $(sinx + cosecx)^2 + (cosx + secx)^2 - (tanx + cotx)^2$ wherever defined is equal to

Let $A = \{\theta \in R : (\frac{1}{3} \sin \theta + \frac{2}{3} \cos \theta)^2 = \frac{1}{3} \sin^2 \theta + \frac{2}{3} \cos^2 \theta\}$. Then:

In a right-angled triangle,the hypotenuse is $2 \sqrt{2}$ times the perpendicular drawn from the opposite vertex. Then the other acute angles of the triangle are

If $\sum_{r=1}^{13} \left\{ \frac{1}{\sin \left(\frac{\pi}{4} + (r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4} + \frac{r\pi}{6}\right)} \right\} = a\sqrt{3} + b$,where $a, b \in \mathbb{Z}$,then $a^2 + b^2$ is equal to:

If $\sin x \cosh y = \cos \theta$ and $\cos x \sinh y = \sin \theta$,then $\sin^2 x + \cosh^2 y =$